Two essays from the organ list

These two essays are pretty good introductions for beginners, with fairly balanced presentations of controversial topics:

http://www.albany.edu/piporg-l/tmprment.html
http://www.albany.edu/piporg-l/meantone.html

On Mon, 21 Feb 2000, Daniel Wolf wrote:
> These two essays are pretty good introductions for beginners, with
> fairly balanced presentations of controversial topics:
>
> http://www.albany.edu/piporg-l/tmprment.html
> http://www.albany.edu/piporg-l/meantone.html

They are pretty good. However, there are a couple of errors in the
second which you may wish to let the author (or the list) know about (I
haven't subscribed to PIPORG-L in some time):

> . . . there were other mean-tone procedures known
> in the 16th and 17th centuries, especially by 2/7th comma (in which, I
> believe, the *minor* 3rds are pure and the major 3rds beat), and 1/3rd
> comma.

In fact it is 1/3rd comma in which minor thirds are pure. In 2/7th
comma both types of third are off by 1/7th comma.

> . . . the "circle of 5ths" is incomplete,
> because of the mathematical impossibility of deriving an octave from a
> series of pure 5ths. Thus, when you start on C and tune pure 5ths, you
> end up with a B# that is substantially different from the original C.
> The difference between the C and B# is called the syntonic comma.

No, this comma is the Pythagorean comma.

--pH <manynote@library.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Do you like to gamble, Eddie?
-\-\-- o Gamble money on pool games?"

Daniel Wolf wrote,

>These two essays are pretty good introductions for beginners, with fairly
balanced presentations of controversial >topics:

<http://www.albany.edu/piporg-l/tmprment.html>
>http://www.albany.edu/piporg-l/tmprment.html
<http://www.albany.edu/piporg-l/meantone.html>
>http://www.albany.edu/piporg-l/meantone.html

I came across the first essay in October, and found a few errors. This was
the response I sent to the author:

************************************************************************
Hello, Stephen --

I very much enjoyed this web page,
http://www.oneskull.dircon.co.uk/3.6.04.htm, and I'm sure many people will
find it useful as an introduction. I thought I caught a few errors, though,
and after consulting with early (esp. Pythagorean) tuning expert Margo
Schulter, have a few corrections to suggest:

Your page says,

>The earliest is PYTHAGOREAN tuning, which seems to have been in use up to
the end of the 16th century. Almost all the >fourths and fifths are dead in
tune, and the entire comma is 'dumped' on one interval (according to Arnaut
de Zwolle between >F and B flat), which is therefore unusable. This tuning
is easy to explain and to execute, but it leaves a lot of the notes of >the
scale in quite odd positions. It is quite satisfactory for music written in
the old 'modes' that preceeded the major and >minor scales, provided there
is no modulation whatever.

Pythagorean tuning gave way to meantone tuning around 1480, and even early
Renaissance music calls for sweet thirds that Pythagorean could not provide.
Except in a few cases:

Margo wrote,

'As far as de Zwolle (c. 1440), that should read "between F# and B" (or,

more strictly, between Gb and B). It's a reference to the tuning with

written sharps placed at the flat end, for certain prominent schisma

thirds, with which we're both acquainted. . . .[As for] tuning with a Wolf
at F-Bb

. . . given that Bb is an integral part of the gamut (_musica recta_), such
a tuning

wouldn't seem too practical for the general repertory. given that Bb is an

integral part of the gamut (_musica recta_), such a tuning wouldn't

seem too practical for the general repertory.'

The schisma thirds are actually diminshed fourths in Pythagorean -- in de
Zwolle's scheme, D-F# (strictly, D-Gb), A-C# (strictly, A-Db), E-G#
(strictly, E-Ab), and B-D# (strictly, B-Eb). These are only a schisma (2
cents) off the perfect 5:4 tuning. Similarly for minor thirds F#-A, C#-E,
and G#-B. So one could almost play music in A major as if it were meantone
using de Zwolle's Pythagorean, except that one would have to avoid the
B-minor triad, as B-F# is the wolf fifth and B-D is Pythagorean. Similarly,
C#-minor works if one avoids the B-major triad. I know major and minor keys
hadn't been invented yet, I'm just trying to get the benefits and
limitations of de Zwolle's system across.

As for modulation, once you've decided to use the schisma thirds, then you
are correct that no modulation is possible -- in fact, one can't even do
everything one wants to in even _one_ key. But the schisma thirds were not
used before 1420, and the "old 'modes'" were simply seven consecutive notes
on the Pythagorean chain. So for hundreds of years through 1420, it would
have been correct to say that the old modes were available with as much
modulational freedom as meantone tuning would subsequently confer on them or
on the major scale.

Also, the major/minor system dates no earlier than about 1600, so music was
written in the "old 'modes'" in the 1500s and yet called for the sweet
thirds of meantone -- Pythagorean would not do. That seems to contradict
your statement above.

Finally you wrote,

>EQUAL TEMPERAMENT. This very obvious solution has been known since 350 BC
(!)

The first correct mathematical description of equal temperament was by
Prince Chu Tsai-y� in 1596, and shorly thereafter in the West. You must be
referring to Aristoxenus, who may have agreed with the concept of equal
temperament (which is controversial) but certainly did not provide any way
of tuning it.

I hope this has been of interest to you and that you will consider revising
your web page.

Thanks,

Paul Erlich